 Okay. Let's get back to where we were. I've got an idea. You know, before you go for a run, you stretch. Make sure you don't get hurt. Let's do a little neurobics. Let's stretch our minds a little bit. On my count, we're going to start at 150, subtract 7, visualize each number and stop at the smallest positive number. Ready? Go. Keep going if you aren't done. Now, here's the thing. There are several levels of that game and they're very important to understand. The lowest level is your lips are moving. 150, 143, 136. Wait a second. Where was I? The second level is your hearing things sort of in your head. 150, 143, 136, 129, 122, 115, etc. That's faster but that's still slow. The next level is your just seeing the answer. It's dead quiet. You see it come up just like a lap counter. And the fourth level, of course, is that you never actually do what you're told. The first thing you always do in any scientific problem or mathematical problem is you try to make it into an easier problem. Subtracting 7 is difficult. So I didn't do that. I subtracted 10 and added 3. 150, 140, 143, 133, 136. Well, pretty soon I can see that the last digit is just going 3, 6, 9, 12, etc, etc. And then I can see all the numbers like one long chain, like a snake. Almost see them all at once down at the bottom. Almost every problem that you encounter is like that in some way. You can get very good at it and if you do practice you will get good at it and if you don't you'll be hopeless. There's a lot of difference between somebody who can do a cartwheel and the rest of us when we try to do a cartwheel. And the real difference is not innate ability, but the person who's practiced it can do it. The person who hasn't practiced it is hopeless. So you've got to practice. Whenever you feel your mind feeling a bit of frustration, you're getting smarter. Okay? When you pressure yourself and you're thinking that's a good feeling. Just don't make it too intense. Okay, let's get on with the work at hand. We're going to talk about partial pressures and the kinetic theory and I've got some topical things. First of all, here's a challenge problem if you're interested. Find the pressure. Remember we did carbon tetrachloride. We got 30.62 liters there. For the volume. So find the pressure for the previous problem so that the van der Waals gas has the same volume as the ideal gas. That's kind of a unique situation that's interesting. And then the second part is find the temperature at which the volume of the van der Waals gas is just barely smaller than the ideal gas. So the pressure's the same and then the final thing is to look up the boiling point of carbon tetrachloride and see if you can think up any reason why you got the answers you did based on how far away we are from the boiling point of the material. Hint. If you're near the boiling point, the gas is not necessarily going to behave that ideally because it's about ready to condense into a liquid. Okay. Intermolecular forces. We talked about van der Waals. We talked about dispersion forces, they're called. Minute charge fluctuations, momentary charge fluctuations leading to the appearance of two dipoles which then attract. But some molecules actually have a permanent dipole. For example, water and chloroform. Because of the difference in the electronegativity of the oxygen and the hydrogen, water has a dipole moment. This side is slightly negative and this side is slightly positive. And chloroform for the same reason, the three big chlorines are big hogs for electrons and they pull all the electrons down this way. And this becomes slightly negatively charged and this becomes slightly positively charged. And both these molecules have relatively high boiling points they persist in the liquid. Water in addition can form hydrogen bonds which is a very strong force compared to the other ones. So that's a special case. But if we have a molecule that has a dipole moment then we expect that it will have a big value of A in the van der Waals formula because it should have a big attractive force because it's a permanent dipole and so they can definitely line up and attract each other under those circumstances. If we compare the role of more electrons which leads to more fluctuations versus a small dipole moment, it turns out that it's a little bit tricky. So if you just say well a dipole is a bigger force than a London dispersion force and I expect this molecule with a dipole to have a bigger value of A than carbon tetrachloride, then you get disappointed because this one that has more electrons actually has the slightly bigger value of A. And this is just the usual case in science. There's a simple cut through which leaves out all the complexity and then there's the real case which is usually more complex and requires a lot more study to figure out why it is that this is bigger than that, that that's smaller than this, and so forth. And that just means that there are different layers like an onion that you have to peel off before you can get very close to reality. If you compare water, water has a smaller value of A but at least in the liquid water would have much higher intermolecular forces than any of the other ones. If you compare H2S which isn't quite as polar as water, H2S is rotten egg gas, very poisonous. We all find H2S extremely unpleasant to smell. If you smell H2S even a little bit you wrinkle your nose and you take off the other way. Why is that? It's very poisonous. The humans who didn't wrinkle their nose and take off the other way are dead. If a volcano erupts and you say boy that smells like hot chocolate and you head toward it, you're gone. Anything that appears in nature that's dangerous like fire or H2S or you name it, you will have an aversion to. But if it doesn't appear in nature all bets are off. HCM you won't find around. That's very poisonous just like H2S but it smells kind of good. And the reason why is nobody ever smelled it so we don't have any kind of selection against that type of thing. If we go down one from sulfur to selenium we can see that as we add more electrons A gets bigger that's comforting. Then we could try going down from carbon. I went all the way to germanium. If we go to germanium tetrachloride then we get this bigger value as well. So there are some trends but it's always more complicated than it seems. Gases aren't very dense. If they were dense they'd be liquids. Sort of by the nature of a gas it's not very dense under normal laboratory conditions. And we can measure the density. We can evacuate a container and then we can flood it with gas and with an accurate balance we can measure the difference in mass. That's not really that big a deal. It's more than you might think too. So if we know the density as the mass per volume and we know the pressure and the temperature and we assume the ideal gas equation then we know how much mass takes up how many moles. But that tells us the molar mass. But just measuring the density of the gas can give us a hint what the gas is. Which one? Not necessarily without some ambiguity because some gases may happen to have the same molar mass like N2 and CO. We wouldn't be able to distinguish them. So we start with PV equals NRT and we divide both sides by V and we divide both sides by RT and then we write the right hand side first. We have the number of moles per volume is equal to the pressure over RT. And the number of moles is the number of grams divided by the molar mass in grams per mole. So I can put that in and then I can solve for the molar mass and the molar mass is the grams RT over PV. Now fine, that's a nice mathematical solution and I do not advise you to use it because the problem with this is the units. The units you're going to have to use are going to be very funny like cubic meters for volume. Okay, so if you convert everything into MKS units you're going to get the answer in kilograms per mole which is not how chemists usually quote it. They usually quote molar mass in grams per mole and you're going to have to use funny units, cubic meters, rs and joules, etc. So here the molar mass is given in kilograms per mole and then only if r is in joules the volume is in cubic meters and the pressure is in these wimpy units pascals, newtons per meter squared. The whole thing is awkward. I don't advise doing it. I'll give you an example problem and I'll show you how I always do it. If the gas is a mixture then the molar mass will come out to be the weighted mean and that might not correspond to any particular gas that you look up in your table of data and you may have to then do some more experiments to figure out what's in there and that's exactly what chemists did to figure out what was in there. Let's burn out all the oxygen. What's left? Let's make this thing react. What's left? They finally had something left that didn't react with anything. That's how they discovered argon. Okay, let's try a practice problem. So let's suppose the density of air is 1.2 grams per liter at one atmosphere and 25 degrees Celsius. What is the apparent molar mass of air? Well, okay. The first thing we always do when we look at these things is look at the units, especially if the units are given in any kind of funny units. I see degree C. I right away think no, no, no. Where is Lord Kelvin? Rolling over in his grave if you use that unit. So we have to convert. And when I convert or when I use a constant, I try to write down as many digits as I can. If I'm given seven digits on a constant on an exam, 8.3144, so I use all those digits. It doesn't take me that long to write them down. People killed themselves to get these constants to that accuracy. It's really a crime to round something off too aggressively and then get the wrong answer. Okay, so go ahead and use 273.15 as the conversion from Celsius to Kelvin. Okay. The way I like to do it then is I like using R in liter atmospheres because I have a liter of gas, I have grams, and I have an atmosphere. And so I just go ahead and write down here 298.15 Kelvin, one atmosphere, one liter. I just solve for the, I say how many number of moles would that be? And I plug in the numbers and I find out that I get 4.087 times 10 to the minus 2 moles. But it said that that was 1.2 grams. So 1.2 grams is 4.07087 times 10 to the minus 2 moles. How many grams per mole? Well, take 1.2 and divide it by that and you end up with 29.36 grams per mole. That's the way I prefer to do it because I find that easier than just using the canned formula and then having messy units all over the place. But you're free to do it any way you want. The only thing that matters is that the orbiter doesn't crash. You have to get the right answer. If you get the right answer, nobody cares how you got it. If you don't get the right answer, nobody cares what great intentions you have. Okay. If you check on your own, you can verify that this squares pretty well with the composition of air that's about 80% nitrogen, about 20% oxygen, and about 1% argon. I've rounded them off. I say others are much smaller but it turns out CO2 is creeping up in fourth place. And as I mentioned, the inert gas argon was in fact discovered by comparing oxygen from air with oxygen from a chemical reaction. But there are certain compounds that when you heat them, they produce oxygen and you prepare that pure compound and heat it and you generate a bunch of oxygen gas. And then you compare the properties of that with oxygen gas that you've isolated from air and you find out that there's a difference. So there's something in the air that's not oxygen that doesn't appear to react anything. They couldn't get it to react. They couldn't find out what it was. So they call it argon, Greek for lazy one. Just doesn't want to react with anything. And in the early days, they called these inert gases because they thought they would never react. Then in the 60s, they got seen on to react with something and then so they called them the noble gases because they would react but not very strongly, sort of like royalty. Okay. So, well, even though argon was inert, chemists knew something else was in there. They just didn't know what it was. If we have mixtures of gases, guess what? They don't care. They all behave independently. So they all just bang against the sides and they behave exactly like the same number of moles of one gas would. And this assumes, of course, that the molecules don't chemically react or do anything, bind to each other in any way. That's what we're talking about. So if we have a mixture of nitrogen and oxygen and argon, that behaves very similarly to the same number of moles of just nitrogen. Not much different. And therefore, we have this concept of partial pressure. The partial pressure of a gas is the pressure due to just that gas alone. Just throw away the other guys. Say, how many moles of this gas do I have? What's the temperature and volume? Whatever that pressure is, that's the partial pressure of the gas. It's a simple idea. Throw away the other guys and there you go. Now, here's a practice problem. What's the partial pressure of oxygen in dry air at STP? STP is an abbreviation for standard temperature and pressure. So STP is 273.15 Kelvin and one atmosphere. And if you actually take the time to look up the detailed composition of air, it's 78.084 percent nitrogen, 20.946 percent oxygen, 0.934 percent argon, and 0.039 percent CO2 due to us adding CO2. It wasn't quite that high before. These are all by volume. But by volume, as long as we treat them as ideal gases, the volume fraction and the mole fraction are the same. We don't have to worry. The mole fraction is just the moles of O2 in the sample divided by the total number of moles of gas in the sample. And it's usually given this Greek symbol chi. The mole fraction of oxygen is the number of moles of oxygen divided by the total. And, well, the partial pressure is just the mole fraction times the atmospheric pressure. The mole fraction is 20 percent. So the partial pressure is 20 percent of one atmosphere or 0.2095 atmospheres. That's the partial pressure of oxygen. And more generally, if we have a ton of different kinds of gases, the partial pressure of any particular gas is the number of moles of that gas divided by the total number of moles in the container times the total pressure, which, if the container is not rigid, is just the external pressure. And we can write that as just Na over Nt times Pt, or we can write this as the mole fraction, this divided by that as the fraction. So that's chi times the total pressure. And that is a mathematical statement of Dalton's law of partial pressure. And in English, we just say the partial pressure of a gas in a mixture is equal to its mole fraction multiplied by the total pressure. It's good to be able to understand it both ways, just as a compact equation and then as a sentence that you can explain to someone else. Okay, let's drive a car. Problem six. A car running under light duty cruise, what does that mean? That means I'm driving very conservatively, five miles under the speed limit and never accelerating unnecessarily. And watching the cars go around you as if there's some great hurry to get home and watch TV because you aren't going exactly at the speed limit. The speed limit's a limit. It's not a suggested rate of travel. It's the limit. If conditions aren't perfect, you don't want to go necessarily at the limit. But let's forget that for now. Let's assume our engine is burning iso-octane, which is C8H18, using air, intake air at atmospheric pressure, iso-octane into a cylinder, compress it and detonate it, make an explosion and then convert that into mechanical energy. Very inefficient, but there you go. So let's assume ideal combustion. The question is, what should the partial pressure of the fuel vapor be in the cylinder? Well, okay. We know the composition of air. It's about 20.946% O2. And let's assume it behaves ideally as an ideal gas. What we need to know then is how much oxygen it takes to burn the iso-octane. Assuming ideal combustion is kind of a code for balance it all to CO2 and H2O. Complete or ideal combustion, hydrocarbon goes to CO2 and H2O. Does it actually do that in a real car? Uh-uh. Usually produces carbon monoxide, very poisonous, and nitrogen oxides called NOx, which give you the brown haze over Los Angeles. But let's ignore that for now. We have iso-octane C8H18 plus some number of moles of O2 unknown, giving some number of moles of CO2 unknown plus some number of moles of H2O unknown. Well, the way you do this kind of thing is you first work with the ones that have to match. There's only two molecules with carbon, this one and this one. This one has eight carbons, this one has one carbon, so Y is eight. That's the first thing you do. Then you look at the hydrogen. There's only two molecules with hydrogen. There's this one with 18, and there's this one with 2, and therefore Z is 9. Once you have Y and Z, you put them in. Now you figure out how much oxygen it takes to balance the oxygen, because oxygen is in both the other guys. You don't start with oxygen, you go around in a circle. We balance the oxygen last, and it comes out to be 25 over 2, 16, and 9. This is molecular oxygen, which is 2, so instead of 25, it's 25 over 2, or 12.5. One thing to notice is that for every mole of iso-octane or gasoline that we burn, we get eight moles of CO2, which is quite a lot. If we want to burn this at the stoichiometric mixture, for every mole of iso-octane, we're going to have to have 12.5 moles of oxygen, but we aren't burning the iso-octane with oxygen, we're burning it with air, and so we've got to intake a ton of nitrogen and all those other things to get enough oxygen into the cylinder to make it work. One thing you might ask yourself after looking at this is, why don't we just burn the fuel with pure oxygen? That would have a lot of great effects. We wouldn't get any NOx because we wouldn't have any nitrogen in the cylinder. It would burn hotter, for sure. Pure oxygen makes things burn extremely hot, but then we need a tank of oxygen to carry around, and we'd have to isolate oxygen from air, which would be energy-intensive. So the short answer is the reason we don't do that is the reason we don't pave the streets with gold. It's just too expensive. It looks nice, it works great, but it's too expensive to do it, so it's not practical. If we burn air, let's figure out how much we need. Well, we've got to have 12.5 moles of oxygen, but oxygen's only 20.946% in air, and so instead of 12.5 moles of oxygen, we need 59.68 moles of air per mole of fuel. So the total number of moles in there is 59.68 plus 1 mole of fuel, so the mole fraction of fuel is 0.01648, or a little more than 1.5% mole percent fuel. When we detonate it, it's mostly air and there's a little fuel that kind of squares with how you imagine it might be in an internal combustion engine. If you think about it, you don't think that it's going to be 50% fuel or anything like that. And if we assume before the compression and detonation that we're bringing stuff in at atmospheric pressure, in other words, we don't have a supercharger or a turbocharger on it, so we aren't actually compressing the gas before we put it in, which would give us more power, perhaps, but maybe a shorter engine life depending how you drive the thing. I think the early subs had a lot of problems. Then the total pressure is one atmosphere, the mole fraction of fuel is 1.6 times 10 to the minus 2, and so it's 1.6 times 10 to the minus 2 atmosphere is the partial pressure of fuel. Final answer. When you tromp on it, the computer controlling the fuel-air mixture doesn't look at the exhaust gas the same way. When you drive conservatively, it tries to adjust it so that there's no O2 left over, but no extra fuel either. That being the most efficient way to go. But if you want more power, so you just say forget efficiency, I want to go, then you put fuel-rich mixture in. And the reason why you do that is so that you have some fuel on the way out to make sure that the cylinder doesn't overheat and just blow up or melt. And then your efficiency goes way down. It would be very illustrative for all of you to ride one time with the guy who drives the car to get the estimated fuel mileage for the car for city and highway driving, those numbers that you never make, 36 and 22 or whatever they are, miles per gallon. They drive the car extremely conservatively. Obviously, if they drive the car very aggressively, the manufacturer goes crazy, says, what are you doing? You're driving my car very aggressively and making me look bad on mileage. So the guy drives the car very, very conservatively, more conservatively than most of you have ever thought of driving. You hardly feel a shift if it's an automatic. Very, very gently. Never push on the accelerator when you don't need to. And then you actually do get that mileage. If you drive aggressively, you also give us worse air quality too because you have fuel-rich mixture coming out. And anybody who runs in the morning can tell whether a car has been driving by in the last two minutes right away. I can tell. No cars on this street. Somebody drove by a little while ago, even if they're gone because there's a difference in air quality. Okay, let's talk about the kinetic theory. We have these empirical laws. A law is a generalization of observations. And a theory is a rationale to explain why that is. And this is a kinetic theory which is designed to explain those laws. We make some assumptions in any theory. And the assumptions we're going to make are four. First, molecules are separated in space in the gas by many diameters. So it is a gas. It's nowhere, nothing like a liquid. Number two, the gas molecules collide and move randomly. That's pretty much fine. After all, gas molecules don't have any brains. So they can't plan what they're going to do. Interestingly, no molecules have any brains. And yet, big assemblages of molecules do have brains or are brains. Number three, there are no attractive or repulsive forces. Whoa! We know that's wrong from Van der Waal. Number four, the average kinetic energy of gas molecules is proportional to the temperature. That one's actually pretty good, in fact, and would follow from number three. Know these assumptions. It's always important to know what the assumptions of a theory are because if any of the assumptions are violated, then the conclusions may not follow. It's very important to ask people what their assumptions are when they forecast an economic model. What are they assuming? And how plausible is that assumption? Or can you verify that that would actually have to happen? Assumption one, well, assumption one is essentially that the molecules have no volume themselves. And that's not going to be true at very high pressure, but this theory is for the theory of gases, gases in the lab, not gases at very high pressure. Assumption number two is fairly accurate. There should be no chemistry occurring, no chemical reactions. They just move randomly. They don't collide and recombine and stick. Assumption number three, as I mentioned, is clearly wrong because we know that all gases can liquefy and that all gases have repulsive forces, too, from the hydraulic jack. But it's mostly okay for traditional gases because we don't try to lift a car with compressed gas. We use a fluid, a liquid, to lift a car. And we're going to talk about gases. And assumption three leads to Dalton's law, in fact. Assumption four is true if there are no forces. If there are no forces, then basically the temperature, the kinetic energy, is the only kind of energy, and that's proportional to the temperature. Potential energy is energy due to position, due to some coordinate. Kinetic energy is energy due to motion. Potential energy is energy like a spring being compressed. But the spring has a force. It's fighting me. If the molecules have no forces between them, there's no springs we can be stretching or otherwise. And so there's no potential energy at all. And therefore the only energy is just the energy of things slamming around. Kinetic energy. And that will be proportional to the temperature. The pressure arises from collisions between gas molecules and the container walls. Gas molecule hits, ricochets, hits here, keeps going boom, boom, boom, boom, boom, boom. Round and round it goes, and that tends to push the thing out. That pushes the pressure out. The gas is trying to escape. It would like to keep going in a straight line. If you want to deflect it, you have to put pressure on it. Make it change direction. If we half the volume, and these are moving randomly, then the number density, the number per unit volume, doubles because we have the same number, half the volume. So we have twice the collisions with the container. But then we're going to get twice the pressure. So if we half the volume, we get twice the pressure based on this little theory. And that was Boyle's law. So this theory explains the empirical observation that Boyle made that he generalized to Boyle's law. That means this theory is on the right track because it's agreeing with experiment. If we increase the temperature, we increase the average kinetic energy in proportion. And with a little bit of math, which I won't go through, we can show that the pressure increases in proportion. So if we increase the temperature, double the absolute temperature, we will double the pressure. But we know that if we keep the pressure constant, the volume would have to double instead. And so if we double the temperature and keep the pressure constant, we will double the volume that means the volume increases linearly with temperature and that's Charles's law. We derive Charles's law from this simple theory as well. And we can also derive Avogadro's law from these assumptions as well. The pressure depends on the number of particles, but not the kind of particles. It's proportional to the number of moles, but not the kind of material. Just the number of particles or number of moles in there. Okay, suppose now we ask a more detailed question because now we're proposing this theory, kinetic theory, theory of motion. The question is, what does it look like in there? I'm a gas molecule. Whoa, I'm flying around. Bang, ow, ricochet off. Boom, boom, two guys. Now I'm stopped, but I can't be stopped for very long because I get hit by a freight train from behind. Oh, and I'm off again and then I may get hit by two other guys and then I'm going really fast. And so the question really is, if we could take a snapshot and stop, look at them all and say, okay, you, how fast are you going? How many meters per second? You, you, you. And we make a histogram. What does it look like? Clearly they aren't going all at exactly the same speed. That's very unlikely. And they can't all be stopped either or that'd be a solid. And they can't all be going really fast or the gas would be hotter than it is. So there has to be some magic distribution which is the most likely, sort of like a curve for a class, the most likely outcome. And Maxwell figured out how to make this histogram and derived the function. So very few molecules are moving slowly. You just can't move slowly in a gas because you get hit by somebody and then you just have to speed up. And you can't be going too fast or too many of them are going too fast, the gas would be hotter than the actual observation temperature. If we imagine suddenly that the gas molecules start moving faster and faster and faster and faster, where did all that energy come from to make them do that? Heat, but we aren't heating it. We're keeping them at constant temperature. Right? Well, no, what I'm saying is that can't happen. If we did heat it, it would happen. We'd get more in the high tail for sure. But if we just sit there with a container at constant temperature and pressure, certain volume, and just say, look, let's take a snapshot. Let's look at what happened. We wouldn't expect to have too many going too fast. Oh, there's a magic distribution that Maxwell worked out. He only lived to be 49. I don't know when this picture was taken, but I think I'm doing better than Maxwell. He liked to drink whiskey. Don't know if that had anything to do with his early demise or not. But he was one of the smartest guys so far. No question about that. And he figured out that it would be this form, the histogram, the number of molecules moving at a speed v, is this spinach under this radical, times v squared times an exponential function which looks like the kinetic energy, mv squared, one-half mv squared, divided by k times t, kb for Boltzmann times t, which is a measure of the thermal energy. So we divide the kinetic energy by the thermal energy. We get a dimensionless thing. Anything in an exponential can't have units because an exponential is a power series, right? 1 plus x plus x squared over 2 factorial, and we can't add up things with different units. And here m is the mass of the particle, and kb is Boltzmann's constant, and t is the absolute temperature in Kelvin. We never use any other temperature. This derivation is really beautiful. And until you see beautiful thought, you may appreciate beautiful art, but you may not yet appreciate beautiful thought because maybe you haven't seen much. But this really is beautiful, but it's a little difficult so it's beyond the scope of our course. This is why you take other courses so you can get to the bottom of certain things like this. It turns out that there's a relationship between the gas constant and Boltzmann's constant, and they're just related by Boltzmann's constant is per molecule, and the gas constant is per mole. So they're related by Avogadro's number. It's a very big 6 times 10 out of 23. And we can then rewrite Maxwell instead of using k, we can use r, and the Avogadro's number stuff cancels out here, and then we get a more useful thing where we have this factor here, which is unfortunately again kilograms per mole, v squared, and then kilograms per mole v squared over 2RT. r has to be in joules here. This is again a slightly tricky formula units wise, and I'll go through a couple of problems to make sure that you understand how to handle the units. So here the molar mass is in kilograms, and r must be in joules, and then you get meters per second for the velocity. Well with this histogram, whenever you have a histogram, if I have a curve for the class, I can say what is the highest grade, what's the lowest grade, what's the average grade, what's the median grade, where half the class is above and half is below that value, and what's the most probable grade? Most people got a 60. I can ask all those things, and the mean, the median, and the most probable or the root mean square, these are statistical measures that tell you about the shape of the distribution. These are used in statistics for any kind of distribution. So let's talk about the mean, the root mean square, and the most probable. Those are ones that we have formulas for that are easy to derive. First of all, the property that this distribution has to have is that the probability of a student having some score has to be one, and that means that the integral of this function has to be unity. Or all the numbers in the histogram have to add up to the total number of people in the class, otherwise I've missed a person, and it obviously can't be more. Now we take the speed here up to infinity, but of course particles can't go faster than the speed of light as far as we know. But our gas particles are nowhere near the speed of light, so that turns out to be a tiny error since there are no particles up there anyway. We can figure out the average speed. The average of anything is the sum over the histogram of the thing. So in the case of a score, you just take the score times how many people got that score. That's all we're doing here. We're taking the velocity times how many molecules have that velocity, what fraction. And we're adding them all up. And that integral, which is easy to do, but I won't do it, turns out to be the square root of 8 RT over pi M. M is in kilograms per mol again. So that's a simple form for the average velocity or the average speed, excuse me, in a gas at a temperature T. We can get the root mean square speed, literally root mean square. Take the square, take the mean of the square, take the square root of the mean of the square, and you get the root mean square. And that's a different integral to do, but it turns out you can do that one too. And you get square root of 3 RT upon M. No pi this time. The other at 8 and pi. And you can get the most probable speed. That's where the slope of the histogram is 0. I'll show you on a plot where that is. And that's easy to do, that requires a little algebra again. And you get the square root of 2 RT over M. And that gives us all our distributions. And here is a picture of what the distributions might look like in meters per second. If I have a heavy molecule like xenon, then I have a much, much slower, much more peak near low speed. And if I have a very light atom, excuse me, not molecule like helium, I find that I get a very wide distribution of values and that the the mean speed is much higher. This is the most probable speed in each case for each of these guys. And you can see that it's not quite symmetrical. There are more guys going fast than there are going slow. So it kind of stretches on the way out. That's the V squared part of that function. These are very fast speeds though. A thousand meters per second. Think of that. That's a kilometer per second. You would guess that we'd get pushed aside or that the air would seem very violent. And it doesn't. In fact, it doesn't do anything. The reason why it doesn't really do anything is because it's like a traffic jam on the 405. Only the cars are actually crashing into each other and that's why they're going nowhere. So I'm going a thousand meters per second but I'm not going in the same direction for very long before I hit something. And then I'm going this way. Then I'm going that way. And on average, I'm going absolutely nowhere unless there's wind, unless there's an actual macroscopic pressure difference somewhere. On the other hand, I do spread out. But we do know that if we put a container here that has perfume in it, eventually you'll start smelling it. In fact, if you cut a rose like Mr. Lincoln would be a good one and you just put it in a vase in your living room on the way out of the door in the morning, when you come back and open the door, the whole house has just a faint beautiful, rich smell of that wonderful flower. So it's diffused during that time throughout the whole place. And you can change the kind of flower every day depending how you feel. You can have one that's a little more citrusy, more cinnamony, little chocolate, et cetera. Okay, let's calculate some relative speeds here. Let's compare the most probable speed for molecular oxygen O2 and carbon tetra bromide CBR4 at 100 degrees Celsius. Ooh, Celsius. Never use Celsius. Well, let's have a look. The most probable speed's the top. We had a formula for it. The most probable speed is the square root of 2RT over M. And here I just wrote out all the units for molecular oxygen 32 grams per mole. I don't want grams because I don't have any grams up here that I can see, convert to kilograms. R, Joules per mole per Kelvin. Kelvin go away. Joules, moles go away. And then I just leave it. I work out the numbers and I get 440 Joules per kilogram to the square root, one-half power. That should be a speed, presumably, if I've done it right. But maybe it's not obvious that it is a speed. But I like to go through this so that I can show you how you can convert it to a speed. Sorry. A joule is a Newton meter because a joule at work is force times distance. And a Newton force is mass times acceleration. Kilograms times meter per second squared. So a Newton times a meter is a kilogram times a meter squared per second squared. But we had a joule per kilogram to the one-half power. So we had meters squared per second squared. The kilograms go away. Square root of meter squared per second squared, meters per second. Sweet. But the answer is, in fact, 440 meters per second since a joule over a kilogram to the one-half power is, in fact, meters per second. On the other hand, for carbon tetra bromide, we plug in carbon 12.01 grams per mole, bromine 79.9 grams per mole. Convert to kilograms. Again, we're going to get joules per kilogram to the one-half. We get 136. Molecular oxygen is moving a lot faster than carbon tetra bromide because it's a lot lighter. And the speeds go like the inverse square root of the molecular masses. Obvious since there it is. Inverse square root. Temperature is the same. That's how it works. And this dependence on the inverse square root explains both diffusion and effusion, both of which can be important. The gas molecules move at high speed, but they don't go anywhere because they keep changing directions. They're like somebody who switches majors three times. They don't get anywhere. Better to select a direction and then go at a moderate rate than go like crazy and then change and reverse gears. Therefore, the net flow of one kind of gas through another one is much, much slower than the individual speeds. And this is called diffusion. Diffusion is the random spreading out of molecules until their concentration everywhere is the same. Whatever molecule is responsible for the fragrance in Mr. Lincoln, and actually there's more than one, they have a certain molar mass and they diffuse out of the petals of the flower as it heats up. And then they spread around the room to some concentration, which is quite small, but your nose can pick it up easily since your nose is a great detector. In liquids, if I put a drop of dye and leave it in a glass of water and come back the next day, the whole water will be a dilute color. The dye will have just randomly spread around. And if I take a pollutant and I dump it in the back bay, it'll randomly spread around everywhere. And then it'll be a million times more expensive to clean it up than if I had caught it early and just scooped it up and gotten it out of there. Once you let things spread out, you're in trouble. It takes a lot of work then to put the genie back in the lamp. You don't want to do that, especially with pollutants. You want to control them right away. Effusion is similar, but it's the escape of a gas from a container with a pinhole into an area of lower pressure like a vacuum, like this. Oop, that guy went out. That one red guy, he went out. And the big heavy silver guys didn't do anything because they're moving too slowly. They just don't hit the sides as often, so they aren't going to get out as easily. Whoever's moving fast goes, he finds a pinhole. Whoever's moving slowly takes forever to get out. But then chemists found they could use this as a way to separate things. If you have a series of pinholes, you eventually get all the fast guys going through. And if they're fast guys or what you want, you have a process to prepare pure fast guys. And the most important aspect of this is in the nuclear fuel cycle. Effusion can be used for isotope separation. We can't separate uranium 235 from uranium 238 by chemistry because they have the same chemistry. Chemistry is about electrons. Isotopes have different numbers of neutrons in the nucleus, but they have the same electron configuration, and so they react pretty much the same. We aren't going to be able to separate these by chemistry. But they have different molar mass. One is 235, one is 238, and therefore they have different rates of effusion. Therefore, if I can prepare a gaseous uranium compound and put it through a ton of pinholes, the U-235 compound will come through the pinholes first. Then I can gather that up and make a bomb, which was the purpose of doing it. Natural uranium is just 0.7 percent U-235. If we had four stages of effusion using the hexafluoride UF6, which turns out to be a gas, what kind of enrichment would we get? Okay. The rate of escape through the pinhole will be like the speed, and that's the inverse square root of the masses. So what we need are the masses, not of just the bare uranium atoms, but of the hexafluorides, which are the gases. So we take the ore, we fluorinate it, we put it into this diffusion apparatus, and we find that one of them is 349 grams per mole, one's 352 grams per mole. Therefore, the rate of effusion of the 235 isomer versus 238 is the inverse, so the 238's on the top, the 235's on the bottom, square root of 352 over 349, 1.0043. That is a miserable number, because that didn't change things much. Well, if you do it four times, you get a factor of 1.0043 each time, so you multiply them, and so if you have four stages of effusion, you now get an enrichment by a factor of 1.0173. Well, if you start with natural ore, which is 0.7 percent, and you get this factor, now your enrichment is 0.712 percent, so it's a little higher. In order to make fuel for a reactor like San Onofre, you have to get the uranium content up to about 2 to 3 percent, say 3 percent uranium 235. That's enough. In order to make a bomb, you need more than 90 percent. So you can see if you're enrich uranium to make a bomb, you're going to have to have a lot of stages of effusion or something in order to do that, and that's usually a big thing that Google Earth can identify and say, hey, what's that thing there that wasn't there before? To make what's called weapons-grade uranium, we need 90 percent. If you have 90 percent and you have eight kilograms, you have a very dangerous thing. Why don't you figure out how many stages of effusion it would take to make a bomb? See, it's going to be a lot. I'll just make a note that centrifuges, which is what the Iranians are using right now, are much more efficient than passive effusion and diffusion technology. With a centrifuge, I put the gases in there, and I just spin it like crazy, and the U-238 goes out more toward the edge, and the U-235 is more toward the middle, and then I suck up a column out of the middle, and then I pump it over to another centrifuge, spin the 238 out, suck up the 235, keep going, keep going, keep going, and at the end, it's pure U-235. Because you have to make it into a gas, if it's uranium metal, it just will sit there like a lump. You could make it into any gas, but most metals don't make gases. It's very hard to make most metals make any kind of a gas. You can make nickel tetra carbonyl. You can make uranium hexafluoride. There are a few examples, but usually it's hard to get a metal atom to go into the gas phase. They like to be solid. Problem with a solid is nothing moves, and so it doesn't effuse. It just sits there, and you can't separate anything. And if you melt it into a liquid, again, you can't really separate anything. You need to get it into the gas phase in order to separate it. That's why you make it. Without 3% enrichment, you just can't produce power. The uranium-235 is the one that produces the power by nuclear fission. U-238 just sits there like wet newspaper. So if you want to light a campfire, you've got to have some kindling and some matches and stuff that burns. And then if you have some plenty of stuff that burns, you can have some wet newspaper hanging around. It won't put it out. If you have too much stuff that burns, like a giant can of gasoline and you light it, then it blows up. So that's why we bring it up to about 3% to generate power. Our own reactor here, because we don't want to have so much uranium, we make it higher percentage. 20% is called low-enriched uranium. Low-enriched uranium is not considered a hazard for anybody making a bomb, because it's still too far away from 90%. It's no worse than ore. If you're a country and you're only interested in power and or research reactors, then you can obtain fuel. We'll sell you the fuel. In fact, we offered to sell the fuel to Iran and they refused. They said, no, we want to enrich to make our own fuel. The problem with building an enrichment plant is that you can keep going and that's why we're a little bit suspicious about what their intentions might be.